The Diophantine equation a(n) = m has a solution (for m >= 1) if and only if floor(arcsinh(sqrt(5)*m/2)/log(phi)) <> floor(arccosh(sqrt(5)*m/2)/log(phi)) where phi is the golden ratio. An equivalent condition is A130259(m) = A130260(m). - Hieronymus Fischer, May 25 2007

a(n+1)= AB^(n)(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 1=`1`, 3=`10`, 8=`100`, 21=`1000`,..., in Wythoff code.

a(n) is also the number of idempotent order-preserving partial transformations (of an n-element chain) of width n (width(alpha) = max(Im(alpha))). Equivalently, it is the number of idempotent order-preserving full transformations (of an n-element chain). - Abdullahi Umar, Sep 08 2008

a(n) is the number of ways that a string of 0,1 and 2 of size (n-1) can be arranged with no 12-pairs. Here a(1)=3, a(2)=8, a(3)=21 and so on. a(n)=1/sqrt(5){phi^(2*n+2)-phi^(-2*n-2)}, where phi=(1+sqrt(5))/2 - The Golden Ratio. - Udita Katugampola, Sep 24 2008

Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n>0, with exactly 2 elements of each rank level above 0. (Uniform used in the sense of Retakh, Serconek, and Wilson. Graded used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 13 2012

a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 3's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011

a(n), n>1 is equal to the determinant of an (n-x) X (n-1) tridiagonal matrix with 3's in the main diagonal, 1's in the super and subdiagonals, and the rest zeros. - Gary W. Adamson, Jun 27 2011